Bass–Quillen conjecture

In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring . The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture.[1][2]

Statement of the conjecture

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The conjecture is a statement about finitely generated projective modules. Such modules are also referred to as vector bundles. For a ring A, the set of isomorphism classes of vector bundles over A of rank r is denoted by  .

The conjecture asserts that for a regular Noetherian ring A the assignment

 

yields a bijection

 

Known cases

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If A = k is a field, the Bass–Quillen conjecture asserts that any projective module over   is free. This question was raised by Jean-Pierre Serre and was later proved by Quillen and Suslin; see Quillen–Suslin theorem. More generally, the conjecture was shown by Lindel (1981) in the case that A is a smooth algebra over a field k. Further known cases are reviewed in Lam (2006).

Extensions

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The set of isomorphism classes of vector bundles of rank r over A can also be identified with the nonabelian cohomology group

 

Positive results about the homotopy invariance of

 

of isotropic reductive groups G have been obtained by Asok, Hoyois & Wendt (2018) by means of A1 homotopy theory.

References

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  1. ^ Bass, H. (1973), Some problems in 'classical' algebraic K-theory. Algebraic K-Theory II, Berlin-Heidelberg-New York: Springer-Verlag, Section 4.1
  2. ^ Quillen, D. (1976), "Projective modules over polynomial rings", Invent. Math., 36: 167–171, Bibcode:1976InMat..36..167Q, doi:10.1007/bf01390008, S2CID 119678534